To find the average rate of change of the function $f(x) = x^2 + 2x + 5$ over the interval $[-4, a]$, we use the formula for the average rate of change between two points $x_1$ and $x_2$ on a function $f(x)$:

$\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

Here:

• $x_1 = -4$
• $x_2 = a$

The formula becomes:

$\text{Average Rate of Change} = \frac{f(a) - f(-4)}{a - (-4)} = \frac{f(a) - f(-4)}{a + 4}$

Step 1: Calculate $f(-4)$

$f(-4) = (-4)^2 + 2(-4) + 5 = 16 - 8 + 5 = 13$

Step 2: Calculate $f(a)$

$f(a) = a^2 + 2a + 5$

Step 3: Plug in the values into the formula

$\text{Average Rate of Change} = \frac{a^2 + 2a + 5 - 13}{a + 4} = \frac{a^2 + 2a - 8}{a + 4}$

So, the average rate of change of the function $f(x) = x^2 + 2x + 5$ over the interval $[-4, a]$ is:

$\boxed{\frac{a^2 + 2a - 8}{a + 4}}$

Do you want any further details or explanations?

Here are 5 related questions:

1. How do you calculate the rate of change for linear functions?
2. What happens to the average rate of change as $a$ increases?
3. How does the function's derivative relate to its average rate of change?
4. Can we simplify the expression further for specific values of $a$?
5. What is the interpretation of the average rate of change for non-linear functions?

Tip: The average rate of change over an interval gives the slope of the secant line through the points on the curve at the interval endpoints.